With three muscles encapsulating the Triceps (tri=3), referring to the “outside of the triceps” is a curious question. By outside, if you mean not the back end of the arm, then the muscle you’re focusing on is the Lateral Head. Single arm reverse pull downs is a great way to target that muscle as well as keeping the elbow tendon from any possible inflammation. Many clients who peform supine (lying down on back) “nose busters” as they’re often called, use too heavy a weight, and flare out their elbows causing excess strain on that tendon.
By standing and using either a cable machine, free motion machine, or resistance band/tubing, you can keep a more stable arm and focus on the triceps. Use a reverse grip (underhand) on a cable handle, elbow bent, hand at chest height. Pull the cable downward, whilc keeping the arm tucked in close the body. Squeeze the muscle at the bottom of the motion (when arm is straight down at side), then slowly (fighting the resistance), bring hand back to starting position. 10-15 reps x 4 sets will fatigue muscle for proper toning/hypertrophy.
Since the triceps is a two joint muscle, it is not only the prime mover for elbow extension, it is also involved in shoulder extension in the sagittal plane to full ROM past the mid-frontal plane.
Lying supine on a bench or on the floor, shoulders flexed to 90 degrees in the sagittal plane, elbows fully flexed, overload only full elbow extension (no movement at the shoulders.) I recommend using dumbells rather than a barbell, since it enables your client to change wrist position from supine to prone. Based on angle of pull, if done with supine wrists, the load shifts toward the lateral head of the triceps. If done with prone wrists, the load shifts toward the medial head.
To overload shoulder extension to full ROM past mid-frontal, I recommend using a machine with double lower cables. Using handles (obviously not a bar!) start with the shoulders adducted in the mid-frontal plane, elbows kept in full extension, overload full ROM shoulder extension in the sagittal plane (n movement at the elbow.) The exercise can be done either standing or seated. It can also be done using dumbells or resistance bands. With supine wrists, load again shifts toward the lateral head, prone wrists load shifts toward the medial head
For most clients I recommend a resistance which leads to temporary muscle failure in 10-15 reps.
There are certainly other exercises, but these should be easily accoplished by most of your clients.